Direct prediction of mid- to far field noise by a CFD calculation is made somewhat impractical because of the meshing and timestep requirements. Firstly, you must solve the time-accurate, compressible Navier-Stokes equations. Accurately resolving the propagation of acoustic waves imposes timestep restrictions such that the Courant number is in the range of 1-2 at most. This restriction can be highly costly. Secondly, the CFD mesh must span all the way to the reception points with enough spatial resolution to directly resolve acoustic waves over the propagation distance with minimal to no numerical damping. These requirements do not make practical sense for many industrial applications.

Practical predications of far field sound pressure levels are made by first starting with a time accurate CFD calculation of the near field region, usually using URANS, LES, DES, SAS or possibly even DNS. Then, assuming that the fluid is a perfect gas, a forced version of the homogeneous wave equation is solved using models for boundary and interior noise sources taken as input from the transient CFD calculation. This solution strategy decouples the source of the noise from the sound propagation and is called the Lighthill, [178] and [179], acoustic analogy.

So, rather than directly predicting far field noise with a time accurate compressible Navier-Stokes solution, you solve for the density fluctuations about a user selected ambient condition:

(15–3)

where is the Lighthill tensor and is the density fluctuation with respect to the ambient condition. Equation 15–3 is the non-homogeneous (driven) wave equation and sometimes called the Lighthill equation. It assumes that the ambient fluid is an isothermal polytropic gas so that the pressure fluctuation is directly proportional to the density fluctuation. As a result, the equation can be written in terms of either pressure or density variation. The Lighthill tensor has three components and is given by:

(15–4)

where the first term is the instantaneous Reynolds stress and is the stress tensor (normal, including pressure, and shear components). Solutions of the Lighthill equation give the spatial and temporal magnitude and distribution of density fluctuations due to interior noise sources generated by the Lighthill tensor. Interior sources result from flow structures such as wakes and shear layers. Surface based sources, not shown in Equation 15–3, can also contribute to density fluctuations and are further discussed in the following section.

One approach to solving the Lighthill equation would be to predict the noise source strength on a separate acoustic mesh, which is normally much coarser than the original CFD mesh, and may cover a much larger spatial extent. This approach significantly reduces the cost of solving for the far field noise over that of a full solution of the compressible Navier-Stokes equations but can still account for complex effects such as reflections, diffraction or absorption by boundary conditions. Other, less computationally expensive approaches are also possible.

Any method that you use to solve the Lighthill equation is a one-way approach that ignores coupling between acoustics and the flow field. Hence, processes that consider this coupling cannot be analyzed with this approach. As a result, the one way methodology is limited to predicting noise mainly for far field regions around aerodynamic devices. Prediction of noise using this approach for a confined, internal flow applications will be of limited usefulness because the flow tends to be more strongly coupled with the acoustics.

Additionally, without even solving the Lighthill equation in some way, it is valuable to evaluate the strength of the various noise sources. This is the approach that can be directly used within CFX to compare designs, as well as to analyze the relative strengths of the noise sources. In addition, the surface noise sources can be written from the flow solver and post processed using software that solves the Lighthill equation.